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A226206
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Number A(n,k) of tilings of a k X n rectangle using integer-sided square tiles of area > 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.
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10
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1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 3, 1, 3, 0, 1, 0, 1, 1, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 1, 1, 0, 1, 1, 5, 0, 7, 0, 5, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 7, 7, 0, 0, 0, 0, 0, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,41
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LINKS
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EXAMPLE
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A(6,4) = A(4,6) = 3:
._._._._._._. ._._._._._._. ._._._._._._.
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|___|___|___| | |___| |___| |
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|___|___|___| |_______|___| |___|_______| .
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...
1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, ...
1, 0, 1, 0, 2, 0, 3, 0, 5, 0, 8, ...
1, 0, 0, 0, 0, 1, 2, 0, 0, 0, 1, ...
1, 0, 1, 1, 3, 2, 7, 7, 16, 19, 40, ...
1, 0, 0, 0, 0, 0, 7, 1, 0, 0, 2, ...
1, 0, 1, 0, 5, 0, 16, 0, 48, 0, 160, ...
1, 0, 0, 1, 0, 0, 19, 0, 0, 50, 17, ...
1, 0, 1, 0, 8, 1, 40, 2, 160, 17, 796, ...
...
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MAPLE
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b:= proc(n, l) option remember; local i, k, s, t;
if max(l[])>n then 0 elif n=0 or l=[] then 1
elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
else for k do if l[k]=0 then break fi od; s:=0;
for i from k+1 to nops(l) while l[i]=0 do s:=s+
b(n, [l[j]$j=1..k-1, 1+i-k$j=k..i, l[j]$j=i+1..nops(l)])
od; s
fi
end:
A:= (n, k)-> `if`(n>=k, b(n, [0$k]), b(k, [0$n])):
seq(seq(A(n, d-n), n=0..d), d=0..14);
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MATHEMATICA
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b[n_, l_List] := b[n, l] = Module[{i, k, s, t}, Which [Max[l] > n, 0, n == 0 || l == {}, 1, Min[l] > 0, t = Min[l]; b[n-t, l-t], True, k = Position[l, 0, 1][[1, 1]]; s = 0; For[i = k+1, i <= Length[l] && l[[i]] == 0, i++, s = s + b[n, Join [l[[1 ;; k-1]], Table[1+i-k, {j, k, i}], l[[i+1 ;; -1]] ]]]; s]]; a [n_, k_] := If[n >= k, b[n, Array[0&, k]], b[k, Array[0&, n]]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 11 2013, translated from Maple *)
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CROSSREFS
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Columns (or rows) k=0-12 give: A000012, A000007, A059841, A079978, A079977, A226369, A226370, A226371, A226372, A226373, A226374, A226375, A226376.
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KEYWORD
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AUTHOR
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STATUS
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approved
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